410 XV NORMED ALGEBRAS AND SPECTRAL THEORY 17. Let E be an infinite-dimensional separable Hilbert space and N a normal continuous operator on E. (a) Show that E is the Hilbert sum EI + E2 of two infinite-dimensional subspaces, each of which is stable under TV and TV*. (Reduce to the case of a simple normal operator M^(lc); with the notation of (15.11.9), observe that if M is a closed subset of Sp(AO, then E(M) can be finite-dimensional only if M is a finite set for which each point has yit-measure ^=0. Then distinguish two cases, according as there exist infinitely many points of measure ^0 or not; in the second case, use Problem 3(b) of Section 13.18.) (b) Deduce from (a) that there exists a decomposition of E as the Hilbert sum of an infinite sequence (En) of infinite-dimensional subspaces, each of which is stable under N and N*. 18. (a) Let E be a separable Hilbert space which is the Hilbert sum of an infinite family (£„)„ 6 z of infinite-dimensional subspaces. There exists a unitary operator S on E such that S(En) = En + i for all n e Z. Show that if P (resp. Q) is the operator which is equal to S1 ~ 2n (resp. S~ 2n) on En for each n, then P2 = Q2 = 1 E , and S = PQ. (b) Deduce from (a) and from Problem 17 that every unitary operator on an infinite-dimensional separable Hilbert space is the product of four involutory unitary operators. (c) Let o> be a complex cube root of unity and let U be the homothety with ratio CD on E, which is a unitary operator. Show that I/ is not the product of three involutory unitary operators. (In general, in a group G, if t is in the center of G, and if there exist x, y, z in G such that / ~ xyz and x2 = y2 — z2 = 1, then also / = yzx, t2 = xyxy and 19. (a) Let E be an infinite-dimensional separable Hilbert space and let (en)n>o be a Hilbert basis of E. The continuous operator K such that V'en = en + 1 for all n^>Qis called the one-sided shift operator; it is an isometry of E onto the hyperplane orthogonal to e0; its spectrum is the disk |£| ^ 1 and contains no eigenvalue of V (Section 11.1, Problem 4); its approximative point-spectrum (Problem 9) is the circle U : |f | = 1. The spectrum of the adjoint operator K* is also the disk|£| <£ 1, and every £ such that |£| < 1 is an eigenvalue of K*. (b) Let T be a continuous operator on E which is an isometry of E onto a (necessarily closed) subspace T(E). Show that there exists a decomposition of E as the Hilbert sum of subspace L and an at most denumerable family (Ft)iei of subspaces, where L and each Ft is stable under T and are such that (1) 7*1 L is unitary and (2) each F( is infinite- dimensional and T\Fi is the one-sided shift operator, for a suitably chosen ortho- normal basis. (Consider the orthogonal supplement N of T(E), and show that E is the Hilbert sum of the TW(N) (n ^ 0) and L = fl Tn(E).) n (c) Deduce from (a) and (b) that if T is any nonunitary isometry of E onto a sub- space of E, then Sp(T) is the unit disk |£| <; 1, and that \\T~ U\\ = 2 for all unitary operators U. (Observe that ||r— U\\ = \\U*T— 1E|| and that U*T is not unitary, so that the point £ = - 1 belongs to its spectrum.) 20. (a) Let E be a separable Hilbert space, T a continuous operator, and C a compact operator on E. Show that the points of Sp(7t+ C) which do not belong to Sp(T) are eigenvalues of T+ C. (Reduce to the case where £ = 0 is such a point and observe that, if T is bijective, we may write T-f- C= T(1E + T~*C), where T~*C is compact; if _i e Sp(7T-1C), it follows that -1 is an eigenvalue of T~1C.)uch that (jitw) converges vaguely to a measure which does not belong to M£(X, v).